We illustrate some of the ideas involved in fractal space-time using f
amiliar deterministic fractals. Starting with the objective of reprodu
cing the Heisenburg uncertainty principle for point particles, we use
the Peano-Moore curve to help visualize the qualitative behaviour of p
articles moving on fractal trajectories in space and time. With this q
ualitative picture in mind we then explore exactly solvable models to
verify that our ideas are mathematically consistent. We find that the
Schrodinger equation describes ensembles of classical particles moving
on fractal random walk trajectories. This shows that the Schrodinger
equation has a straightforward microscopic model which is not, however
, appropriate for quantum mechanics. The free particle Dirac equation
is also derivable in terms of ensembles of classical particles and thi
s unites the two equations conceptually in a very direct way. In both
cases what we discover is a many-particle simulation of quantum mechan
ics and this confirms in a graphic way that the mysteries surrounding
quantum mechanics lie not in the equations, but in interpretation and
the theory of measurement. Finally, we discuss an exactly solvable mod
el which incorporates fractal time. The calculation produces the Dirac
equation in 1 + 1 dimensions and because of intrinsic space-time loop
s, constitutes a model with the potential to exhibit the wave-particle
duality found in nature. Copyright (C) 1996 Elsevier Science Ltd